Bifurcation Diagram

- order & chaos

Bifurcation diagram is plotted from the function f(x) = kx (1-x) for different values of k ( a constant).

We start with k as 1 and x as 0.01. The new value of the function is calculated and plotted on the diagram. The value is then substituted back in the formulae and a new value of x calculated. This is repeated 64 times. The value of k is then increased slightly and the whole process repeated.

When k is less than 2.8 a simple curve is produced. As k increases, the curve splits into two. The formulae has two stable states. At higher values of k, the formulae becomes chaotic, with multiple points being plotted.

To New
  # set default screen, pen and turtle values
  ResetAll SetScreenSize [400 400] HideTurtle
  SetSC Black SetPC Green SetPS 1 PenUp
End

To Init
  # change the start and end for magnified plots
  Make "Start 1 Make "End 4
  Make "Step (:End-:Start)/360
End

To Function :K :X
  Output :K*:X * (1-:X)
End

To Display
  # write header title and footer values
  PenUp SetPC White SetH 0
  SetPos [-190 182] Label [Bifurcation Diagram]
  SetPos [-190 -190] Label ( List "k\ = :Start "to :End )
End

To Go
  New Init Display SetPC Green
  Make "X 0.1 Make "K :Start
  # first iterate to eliminate transients
  Repeat 128 [
    Make "Xn Function :K :X
    Make "X :Xn]
  # actual iteration
  For (List "K :Start :End :Step) w[
    Repeat 64 [
      Make "Xn Function :K :X
      Make "X :Xn
      # calculate horiz x
      Make "Xh (360/(:End-:Start))*:K-180-((360/(:End-:Start))*:Start)
      Dot List :Xh (360*:X)-180] ]
End

Bifurcation
Bifurcation

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